Coarse embeddability of the Hilbert space into Banach spaces

A coarse embedding between metric spaces is, intuitively speaking, a map which preserves, in a weak sense, the geometry at large distances. More precisely, if $(M,d)$ and $(N,\delta)$ are metric spaces, $f:M\to N$ is a coarse embedding if there exist $\rho,\omega:[0,\infty)\to [0,\infty)$, with $\lim_{t\to \infty}\rho(t)=\infty$ and such that for all $x,y$ in $M$, $\rho(d(x,y))\le \delta(f(x),f(y))\le \omega(d(x,y))$.

The question of embedding metric spaces into ``nice'' Banach spaces has gathered researchers coming from very diverse origins. The starting point is to try to get a better understanding of complicated metric spaces, by embedding them into well understood Banach spaces, like Hilbert spaces. For instance, a remarkable result of G. Yu (2000) asserts that the validity of the Novikov conjecture is insured by the coarse embeddability of the Cayley graph of a certain fundamental group into a Hilbert space. Therefore a seemingly weak metric assumption on the group yields a very strong result in differential topology.

On the other hand, many results, including of course Dvoretksy's theorem, indicate that in some sense the Hilbert space $\ell_2$ is the most difficult Banach space to embed into. We will recall them. However, the question to know whether $\ell_2$ coarsely embeds into any infinite dimensional Banach space was left open.

In this talk we will show, by looking at the ``famous'' Tsirelson space, that this is not the case. The other main tool of the proof relies on Nigel Kalton's idea to use fundamental countably branching graphs to study non linear embeddings. This is a joint work with F. Baudier and Th. Schlumprecht.